Graphical Analysis of Fluid Flow Through Polymeric Complex Structures Using Multi-scale Simulations - Selected Topics of Computational and Experimental Fluid Mechanics - page 285

D

r ij )
is a dimensionless weight function. The random force disperses the heat originated
by the dissipative force and converts it into Brownian motion by keeping the local
temperature T constant. It is expressed as:

ʳ

is the dissipation constant, v ij = v i −

ˉ (

where v j is the relative velocity, and

F ij

R

=− ˃ˉ ( r ij ) ʾ ij ˆ e ij (7)

/ √ ʴ t ) with ʾ ij = ʸ ij ( 1 , where ʸ ij is a random Gaussian number with zero mean and

unit variance and ʴ t is the integration time-step. As it has been pointed out before

these two forces are related as r ij ) 2 .

Finally, conservative forces account for local hydrostatic pressure and are of the
form

r ij ) = ˉ

D R

ˉ ( (

a ij ˉ

c

(

r ij ) ˆ e ij ,( r ij < r c )

F ij =

(8)

0 , ( r ij ≥ r c ).

In this equation, a ij is a very important parameter because it represents the maximum
repulsion between particles i and j and a good parametrization of this is essential to
obtain a realistic representation of our systems. In addition, r ij = r i − r j , r ij =| r ij | ,

and ˆ e ij = r ij / r ij , where r i is the position of particle i and the weight function is

c

given by r c . If an adequate parametrization of the a ij value is

stabilized it is possible to obtain a very good representation of the structure of many
complex fluids.

ˉ ( r ij ) = 1 − r ij /

3 Simulations and Visualization

In this section, we present some pictures obtained by the visual study of polymeric
and non-polymeric fluid flow simulations through different porous media using the
Smoothed Particle Hydrodynamics (SPH) and the Dissipative Particle Dynamics
(DPD) methodologies. A complete description of the pictures is given in their cor-
responding figure captions. (see Figs. 1 , 2 , 3 , 4 )